The asymptotic performance of binary- and multilevel-logic distributed decision fusion systems is studied. We define as perfect detectability the condition at which the detection probability at the fusion approaches one as the number of available decisions at the fusion approaches infinity, while the false alarm probability approaches zero. We investigate the performance of the fusion when the sensors transmit to the fusion the output of binary or multi-level memoryless nonlinearities. When the binary or multi-level decisions that the sensors transmit are independent conditioned on either hypothesis, it is shown that, under a Neyman-Pearson fusion rule, perfect detectability is achievable at an exponential rate for similar and dissimilar sensors, provided that the receiver operating characteristic (ROC) of each sensor lies above a lower bound ROC. The lower bound ROC has been calculated numerically for the case of similar sensors. The resulting lower bound ROC is very 'liberal,' in the sense that the ROC of any reasonable detector lies above it.